In the PAC-Bayesian literature, the C-Bound refers to an insightful relation between the risk of a majority vote classifier (under the zero-one loss) and the first two moments of its margin (i.e., the expected margin and the voters' diversity). Until now, learning algorithms developed in this framework minimize the empirical version of the C-Bound, instead of explicit PAC-Bayesian generalization bounds. In this paper, by directly optimizing PAC-Bayesian guarantees on the C-Bound, we derive self-bounding majority vote learning algorithms. Moreover, our algorithms based on gradient descent are scalable and lead to accurate predictors paired with non-vacuous guarantees.
University of Almería（阿尔梅里亚大学）；University of Copenhagen（哥本哈根大学）
We present a novel analysis of the expected risk of weighted majority vote in multiclass classification. The analysis takes correlation of predictions by ensemble members into account and provides a bound that is amenable to efficient minimization, which yields improved weighting for the majority vote. We also provide a specialized version of our bound for binary classification, which allows to exploit additional unlabeled data for tighter risk estimation. In experiments, we apply the bound to improve weighting of trees in random forests and show that, in contrast to the commonly used first or…
University of Copenhagen（哥本哈根大学）；University of Aalborg（奥尔堡大学）
We present a new second-order oracle bound for the expected risk of a weighted majority vote. The bound is based on a novel parametric form of the Chebyshev- Cantelli inequality (a.k.a. one-sided Chebyshev's), which is amenable to efficient minimization. The new form resolves the optimization challenge faced by prior oracle bounds based on the Chebyshev-Cantelli inequality, the C-bounds [Germain et al., 2015], and, at the same time, it improves on the oracle bound based on second order Markov's inequality introduced by Masegosa et al. [2020]. We also derive a new concentration of measure inequ…
Univ Lyon（里昂大学）；UJM-Saint-Etienne（圣埃蒂安让·莫内大学）；CNRS（法国国家科学研究中心）；Institut d Optique Graduate School（光学研究所研究生院）；Université Laval（拉瓦尔大学）；Inria（法国国家信息与自动化研究所）；Inria London Programme（法国国家信息与自动化研究所伦敦项目）；University College London（伦敦大学学院）
We investigate a stochastic counterpart of majority votes over finite ensembles of classifiers, and study its generalization properties. While our approach holds for arbitrary distributions, we instantiate it with Dirichlet distributions: this allows for a closed-form and differentiable expression for the expected risk, which then turns the generalization bound into a tractable training objective. The resulting stochastic majority vote learning algorithm achieves state-of-the-art accuracy and benefits from (non-vacuous) tight generalization bounds, in a series of numerical experiments when com…
Univ Lyon（里昂大学）；UJM-Saint-Etienne（圣艾蒂安大学）；CNRS（法国国家科学研究中心）；Institut d’Optique Graduate School（高等光学研究院）；Univ. Grenoble Alps（格勒诺布尔阿尔卑斯大学）；Inria Lille - Nord Europe（法国里尔-北欧国家信息与自动化研究所）
In this paper we propose a boosting based multiview learning algorithm, referred to as PB-MVBoost, which iteratively learns i) weights over view-specific voters capturing view-specific information; and ii) weights over views by optimizing a PAC-Bayes multiview C-Bound that takes into account the accuracy of view-specific classifiers and the diversity between the views. We derive a generalization bound for this strategy following the PAC-Bayes theory which is a suitable tool to deal with models expressed as weighted combination over a set of voters. Different experiments on three publicly avail…
Université Laval（拉瓦尔大学）；Laboratoire Hubert Curien UMR CNRS 5516（于贝尔·居里安实验室 CNRS UMR 5516）；Univ. Jean Monnet（让·莫内大学）；Aix-Marseille Univ.（艾克斯-马赛大学）；CNRS（法国国家科学研究中心）
Department of Computer Science, University of Copenhagen（哥本哈根大学计算机科学系）
Existing guarantees in terms of rigorous upper bounds on the generalization error for the original random forest algorithm, one of the most frequently used machine learning methods, are unsatisfying. We discuss and evaluate various PAC-Bayesian approaches to derive such bounds. The bounds do not require additional hold-out data, because the out-of-bag samples from the bagging in the training process can be exploited. A random forest predicts by taking a majority vote of an ensemble of decision trees. The first approach is to bound the error of the vote by twice the error of the corresponding G…
In this paper, we propose a diversity-aware ensemble learning based algorithm, referred to as DAMVI, to deal with imbalanced binary classification tasks. Specifically, after learning base classifiers, the algorithm i) increases the weights of positive examples (minority class) which are "hard" to classify with uniformly weighted base classifiers; and ii) then learns weights over base classifiers by optimizing the PAC-Bayesian C-Bound that takes into account the accuracy and diversity between the classifiers. We show efficiency of the proposed approach with respect to state-of-art models on pre…
We present a new concentration of measure inequality for sums of independent bounded random variables, which we name a split-kl inequality. The inequality is particularly well-suited for ternary random variables, which naturally show up in a variety of problems, including analysis of excess losses in classification, analysis of weighted majority votes, and learning with abstention. We demonstrate that for ternary random variables the inequality is simultaneously competitive with the kl inequality, the Empirical Bernstein inequality, and the Unexpected Bernstein inequality, and in certain regim…
In this paper, we consider mixtures of two Mallows models for top-$k$ rankings, both with the same location parameter but with different scale parameters, i.e., a mixture of concentric Mallows models. This situation arises when we have a heterogeneous population of voters formed by two homogeneous populations, one of which is a subpopulation of expert voters while the other includes the non-expert voters. We propose efficient sampling algorithms for Mallows top-$k$ rankings. We show the identifiability of both components, and the learnability of their respective parameters in this setting by, …
Meta learning automatically infers an inductive bias, that includes the hyperparameter of the base-learning algorithm, by observing data from a finite number of related tasks. This paper studies PAC-Bayes bounds on meta generalization gap. The meta-generalization gap comprises two sources of generalization gaps: the environment-level and task-level gaps resulting from observation of a finite number of tasks and data samples per task, respectively. In this paper, by upper bounding arbitrary convex functions, which link the expected and empirical losses at the environment and also per-task level…
We explore the family of methods "PAC-Bayes with Backprop" (PBB) to train probabilistic neural networks by minimizing PAC-Bayes bounds. We present two training objectives, one derived from a previously known PAC-Bayes bound, and a second one derived from a novel PAC-Bayes bound. Both training objectives are evaluated on MNIST and on various UCI data sets. Our experiments show two striking observations: we obtain competitive test set error estimates (~1.4% on MNIST) and at the same time we compute non-vacuous bounds with much tighter values (~2.3% on MNIST) than previous results. These observat…
Reconcile Certified Robustness and Accuracy for DNN-based Smoothed Majority Vote Classifier
National Mobile Communications Research Laboratory, Southeast University, China（东南大学国家移动通信研究中心）；Department of Computer Science, University of Liverpool, UK（利物浦大学计算机科学系）
Univ Lyon, UJM-Saint-Etienne, CNRS, Institut d Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂大学，圣埃蒂安让·莫内大学，法国国家科学研究中心，光学研究生院，于贝尔·居里安实验室 UMR 5516）；Airbus Opération S.A.S（空中客车运营公司）；University of Toulouse, Institut de Recherche en Informatique de Toulouse（图卢兹大学，图卢兹信息研究所）
We propose the first general PAC-Bayesian generalization bounds for adversarial robustness, that estimate, at test time, how much a model will be invariant to imperceptible perturbations in the input. Instead of deriving a worst-case analysis of the risk of a hypothesis over all the possible perturbations, we leverage the PAC-Bayesian framework to bound the averaged risk on the perturbations for majority votes (over the whole class of hypotheses). Our theoretically founded analysis has the advantage to provide general bounds (i) that are valid for any kind of attacks (i.e., the adversarial att…
PAC-Bayesian Generalization Guarantees for Fairness on Stochastic and Deterministic Classifiers
CNRS（法国国家科学研究中心）；Institut d Optique Graduate School（高等光学学院）；Inria（法国国家信息与自动化研究所）；Institut Universitaire de France（法国大学研究院）；Universite Claude Bernard Lyon 1（里昂第一大学）；Univ Rennes（雷恩大学）
Microsoft Research（微软研究院）；University of Illinois at Chicago（伊利诺伊大学芝加哥分校）
In the problem of learning with label proportions, which we call LLP learning, the training data is unlabeled, and only the proportions of examples receiving each label are given. The goal is to learn a hypothesis that predicts the proportions of labels on the distribution underlying the sample. This model of learning is applicable to a wide variety of settings, including predicting the number of votes for candidates in political elections from polls. In this paper, we formally define this class and resolve foundational questions regarding the computational complexity of LLP and characterize i…
A PAC-Bayesian View of Generalisation for Physics-Informed Machine Learning
Université Jean Monnet Saint-Étienne, CNRS, Institut d’Optique Graduate School, Laboratoire Hubert Curien UMR 5516（里昂蒙特大学圣埃蒂安分校、法国国家科学研究中心、光学研究生院、Hubert Curien实验室 UMR 5516）；Inria and University College London, France and United Kingdom（Inria 和 英国伦敦大学学院，法国和英国）
Recursive PAC-Bayes: A Frequentist Approach to Sequential Prior Updates with No Information Loss
University of South Denmark（南丹麦大学）；University of Copenhagen（哥本哈根大学）；Novo Nordisk A/S（诺和诺德公司）；CNRS（法国国家科学研究中心）；Sorbonne Université（索邦大学）；Université Paris Cité（巴黎西岱大学）
University College London（伦敦大学学院）；Inria（法国国家信息与自动化研究所）；ServiceNow Research（ServiceNow研究院）
We study the generalisation properties of majority voting on finite ensembles of classifiers, proving margin-based generalisation bounds via the PAC-Bayes theory. These provide state-of-the-art guarantees on a number of classification tasks. Our central results leverage the Dirichlet posteriors studied recently by Zantedeschi et al. [2021] for training voting classifiers; in contrast to that work our bounds apply to non-randomised votes via the use of margins. Our contributions add perspective to the debate on the "margins theory" proposed by Schapire et al. [1998] for the generalisation of en…
Google Research, Brain Team（谷歌研究院大脑团队）；Toyota Technological Institute at Chicago（芝加哥丰田理工学院）
The classical PAC sample complexity bounds are stated for any Empirical Risk Minimizer (ERM) and contain an extra logarithmic factor $\log(1/ε)$ which is known to be necessary for ERM in general. It has been recently shown by Hanneke (2016) that the optimal sample complexity of PAC learning for any VC class C is achieved by a particular improper learning algorithm, which outputs a specific majority-vote of hypotheses in C. This leaves the question of when this bound can be achieved by proper learning algorithms, which are restricted to always output a hypothesis from C. In this paper we aim to…
We provide two main contributions in PAC-Bayesian theory for domain adaptation where the objective is to learn, from a source distribution, a well-performing majority vote on a different, but related, target distribution. Firstly, we propose an improvement of the previous approach we proposed in Germain et al. (2013), which relies on a novel distribution pseudodistance based on a disagreement averaging, allowing us to derive a new tighter domain adaptation bound for the target risk. While this bound stands in the spirit of common domain adaptation works, we derive a second bound (introduced in…
